Question: The median of the set of numbers $\{$12, 38, 45, $x$, 14$\}$ is five less than the mean. If $x$ is a negative integer, what is the value of $x$?
Since $x$ is negative, the median of the set is 14.  Therefore, the mean of the set is $14+5=19$, and its sum is $19\cdot 5=95$.  Since 12, 38, 45, and 14 sum to 109, the remaining integer $x$ must be $95-109=\boxed{-14}$.